Cardinal Math

Grades 3–5 · 3.NF.3, 4.NF.1, 4.NF.2, 4.NF.3, 5.NF.1

How do I help my child with fractions?

Fractions: equal parts, equivalence, adding

  • A fraction counts pieces of one size. 3/4 is three pieces, each a fourth: the bottom number names the piece, the top counts them.
  • Nearly every fraction mistake comes from treating the two numbers as two ordinary numbers. Add them, compare them, grow them: all three fail.
  • So cut things: paper strips, a chocolate bar, a pizza on a napkin. Fourths have to be equal, and two fourths have to visibly match one half.
  • Find your child's wrong answer below and you've found the rule they're using.

The five-minute check

A short run of problems on a phone or tablet, chosen as your child answers. If one of the five patterns is behind their mistakes, the check names it. Nothing to install, nothing stored.

Hand the phone over, or play it here. When it says all done, take it back and tap see what we found.

Start the five-minute check

The five ways it goes wrong

  1. Adds the tops and adds the bottoms 1/4 + 2/4 they write 3/8 it is 3/4 Adds the numerators and adds the denominators: 1/4 + 2/4 is 3/8.
  2. Adds the same number to top and bottom Equal to 2/3 they write 3/4 it is 4/6 Makes an equivalent fraction by adding to both parts: 2/3 becomes 3/4.
  3. Bigger numbers mean a bigger fraction Which is bigger, 1/3 or 1/8? they write 1/8 it is 1/3 Orders fractions by their numbers: 1/8 is bigger than 1/3 because 8 is bigger than 3.
  4. Judges by the gap to one Which is bigger, 5/6 or 7/8? they write the same it is 7/8 Compares by how many pieces are missing from a whole: 5/6 and 7/8 are both one short, so they must be equal.
  5. Takes the smaller piece from the larger 3 1/4 − 1 3/4 they write 2 2/4 it is 1 2/4 In 3 1/4 − 1 3/4, takes 1/4 from 3/4 because that's the way round that works, and gets 2 2/4.

Why children reach for each one, and what replaces it →

What the school is probably doing

Area models
Circles and bars cut into equal parts and shaded. Still the best picture for equal parts.
Fraction strips
Paper strips folded into halves, fourths, eighths. Where equivalence becomes obvious instead of a rule.
Number lines
A fraction as a point between 0 and 1. Schools lean on this from grade 3 because it makes a fraction a number, not a picture of pizza.
Equivalence by multiplying
Top and bottom multiplied by the same number: folding every piece again, written down.

Questions parents ask

What grade is this?

Equal parts and unit fractions start in grade 3 in the US (3.NF), equivalence and adding with the same denominator are grade 4 (4.NF.1, 4.NF.3), and adding with different denominators is grade 5 (5.NF.1).

Why can't you add the bottoms?

The bottom number isn't a quantity, it's the name of the piece. Two fourths and one fourth are three fourths, the way two apples and one apple are three apples.

Is 2/4 the same as 1/2?

Same amount, different name. Fold a strip in half and shade one half, then fold it again: the shaded part is now two fourths and nothing changed.

Pizza or number line?

Both, in that order. Pizza for equal parts and adding same-size pieces; the number line for seeing 3/4 as a number between 0 and 1, and for comparing.

They say 1/8 is bigger than 1/3.

The bottom number is still a quantity to them. The chocolate bar shared three ways and eight ways settles it fast; the words take longer.

How long should we practise?

Five to ten minutes most days beats an hour on Sunday. Three or four problems done slowly with a folded strip are worth more than a page.

Cardinal Math watches for all of these, every day.

Inside the app the same patterns are caught as your child plays, across the whole of K to grade 5, and explained to you in plain words. One price, no ads, no streaks. Coming to iPhone and iPad.

Tell me when it's out